# Inner Automorphisms

• Mar 8th 2010, 10:46 PM
bweiland
Inner Automorphisms
I'm just having an off night and while reading through an algebra book I came across that center Z(G) is the kernel in this, but I can't verify it to myself. Can anyone verify the reason for this?

This is what I'm talking about: Center (group theory) - Wikipedia, the free encyclopedia
• Mar 8th 2010, 11:09 PM
bweiland
Never mind. I figured it out. It follows since the center is a subgroup and has the property that x ∈Z(G), g∈G then gxg-1∈Z(G).
• Mar 8th 2010, 11:35 PM
aliceinwonderland
Quote:

Originally Posted by bweiland
I'm just having an off night and while reading through an algebra book I came across that center Z(G) is the kernel in this, but I can't verify it to myself. Can anyone verify the reason for this?

This is what I'm talking about: Center (group theory) - Wikipedia, the free encyclopedia

Every element g of a group G, an inner automorphism $\phi_g:G \rightarrow G$ defined by $\phi_g(h)=ghg^{-1}$. Verfiy that $\phi_g$ is an automorphism of G for given $h \in G$. Now, let $f:G \rightarrow \text{Aut(G)}$ defined by $f(g)=\phi_g$.The kernel of this map, which maps to an identity automorphism, is $\{g \in G | \phi_g(h)=h \text{ for all } h \in G\}=\{g \in G | gh=hg \text{ for all } h \in G\}$.
• Mar 9th 2010, 09:29 AM
bweiland
I'm glad you still posted that reply because it reminded me that I've been writing my functions in more of an analysis format, which is just nonsense in algebra.
• Mar 9th 2010, 01:50 PM
Drexel28
Quote:

Originally Posted by bweiland
I'm glad you still posted that reply because it reminded me that I've been writing my functions in more of an analysis format, which is just nonsense in algebra.

What's the analysis form of functions?
• Mar 9th 2010, 02:27 PM
bweiland
I meant that I was writing my functions more like you would in calculus.

So, for a mapping, I should have θ(g)=ιg(x) where ιg(x) = gxg-1 but instead I was writing θ(g) =gxg-1 and that makes no sense since it doesn't define x.
• Mar 9th 2010, 02:29 PM
Drexel28
Quote:

Originally Posted by bweiland
I meant that I was writing my functions more like you would in calculus.

So, for a mapping, I should have θ(g)=ιg(x) where ιg(x) = gxg-1 but instead I was writing θ(g) =gxg-1 and that makes no sense since it doesn't define x.

I would really say that $\theta_g(x)=gxg^{-1}$. The subscript notation denotes that $g$ is fixed upon inspection.

P.S. That is how you write a lot of things in alysis ;)
• Mar 9th 2010, 03:35 PM
bweiland
Exactly, but in algebra it is poor notation.